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A132870
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Triangle T, read by rows, where the g.f. of row n of T^n = (n^2 + y)^n for 0 <= n <= 29, where T^n denotes the n-th power of T considered as (lower-left) matrix.
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11
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1, 1, 1, 6, 4, 1, 132, 45, 9, 1, 7156, 1432, 168, 16, 1, 729895, 101725, 7550, 450, 25, 1, 119636226, 12938076, 697590, 27420, 990, 36, 1, 28619359629, 2559100705, 110137692, 3226895, 78890, 1911, 49, 1, 9374688646296, 721024536688, 26208036736, 624158528, 11572400, 193312, 3360, 64, 1
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OFFSET
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0,4
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COMMENTS
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Pascal's triangle, C, obeys: g.f. of row n of C^n = (n + y)^n for n >= 0.
Starting from row 30 on, the terms computed by the given formulae are not integers any more (as noticed by Alois P. Heinz). - M. F. Hasler, Nov 19 2017
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LINKS
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FORMULA
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Diagonals: T(n,n) = 1 for n >= 0. T(n,n-1) = n^2 for n >= 1. T(n, n-2) = (n-1)^2*n*(2n-1)/2. - M. F. Hasler, Nov 19 2017
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EXAMPLE
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Triangle T begins:
1;
1, 1;
6, 4, 1;
132, 45, 9, 1;
7156, 1432, 168, 16, 1;
729895, 101725, 7550, 450, 25, 1;
119636226, 12938076, 697590, 27420, 990, 36, 1;
28619359629, 2559100705, 110137692, 3226895, 78890, 1911, 49, 1;
9374688646296, 721024536688, 26208036736, 624158528, 11572400, 193312, 3360, 64, 1;
...
Matrix square T^2 (padded with 0's to the right of the diagonal) begins:
1;
2, 1;
16, 8, 1; <== g.f. of row 2: (2^2 + y)^2
363, 126, 18, 1;
18864, 4256, 480, 32, 1;
1845115, 289700, 23350, 1300, 50, 1; ...
Matrix cube T^3 begins:
1;
3, 1;
30, 12, 1;
729, 243, 27, 1; <== g.f. of row 3: (3^2 + y)^3
37380, 9048, 936, 48, 1;
3534210, 614925, 51000, 2550, 75, 1; ...
Matrix 4th power T^4 begins:
1;
4, 1;
48, 16, 1;
1266, 396, 36, 1;
65536, 16384, 1536, 64, 1; <== g.f. of row 4: (4^2 + y)^4
6058330, 1142800, 94100, 4200, 100, 1; ...
Matrix 5th power T^5 begins:
1;
5, 1;
70, 20, 1;
2010, 585, 45, 1;
106740, 26840, 2280, 80, 1;
9765625, 1953125, 156250, 6250, 125, 1; <== g.f. of row 5: (5^2 + y)^5
1431275130, 222982380, 15380550, 601500, 13950, 180, 1; ...
etc.
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MATHEMATICA
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pt = {{1}}; Table[rhs = CoefficientList[(k^2 + x)^k, x];
qt = Join[pt, {vars = Array[Subscript[a, #] &, k + 1]}];
b = MatrixPower[PadRight[qt], k] ;
{out} = vars /. Solve[Thread[Reverse[Last[b]] == Reverse[rhs]], vars];
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PROG
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(PARI) T(n, k=-1/*omit 2nd arg to get the whole table up to row n*/, M=Mat(1))={ for(m=#M, n, M=matid(m+1)-matconcat([M^m; Vecrev(('x+m^2)^m)]); M=sum(i=1, m+1, -M^i/i)/m; M=sum(i=0, m+1, M^i/i!)); if(k>=0, M[n+1, k+1], M)}\\ Rewritten by M. F. Hasler, Nov 19 2017
for(n=0, 10, for(k=0, n, print1(T(n, k), ", ")); print(""))
A=T(29); c=-1; for(n=0, 29, for(k=0, n, write("/tmp/b132870.txt", c++, " ", A[n+1, k+1]))) \\ M. F. Hasler, Nov 18 2017
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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